Another way of showing the harmonic series diverges, without all those fancy comparisons to integrals (and having to know that limx->infinity log(x) = infinity, which is somewhat harder to prove), is to perform the following elementary comparison.
1 >= 1
1/2 >= 1/2
1/3 >= 1/4
1/4 >= 1/4
1/5 >= 1/8
1/6 >= 1/8
1/7 >= 1/8
1/8 >= 1/8
...
Generally,
each of the terms 1/(2
k+1), 1/(2
k+2), ...,
1/2
k+1 is
at least 1/2
k+1, and there are 2
k such terms. Thus, the sum of these terms is
at least 1/2, so the sum of the first 2
k terms of the harmonic series is at least
k/2; thus, the
series diverges.
Well actually, I lied. This sort of thing is called the condensation test for series convergence...